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{{Use American English|date = March 2019}}
{{Short description|Affine connection on the tangent bundle of a manifold}}
In [[Riemannian manifold|Riemannian]] or [[pseudo-Riemannian manifold|pseudo-Riemannian geometry]] (in particular the [[Lorentzian manifold|Lorentzian geometry]] of [[general relativity]]), the '''Levi-Civita connection''' is the unique [[affine connection]] on the [[tangent bundle]] of a [[manifold]] (i.e. [[affine connection]]) that [[metric connection|preserves]] the ([[pseudo-Riemannian manifold|pseudo-]])[[Riemannian metric]] and is [[Torsion (differential geometry)|torsion]]-free.

The [[fundamental theorem of Riemannian geometry]] states that there is a unique connection that satisfies these properties.

In the theory of [[Riemannian manifold|Riemannian]] and [[pseudo-Riemannian manifold]]s the term [[covariant derivative]] is often used for the Levi-Civita connection. The components (structure coefficients) of this connection with respect to a system of local coordinates are called [[Christoffel symbols]].

== History ==
The Levi-Civita connection is named after [[Tullio Levi-Civita]], although originally "discovered" by [[Elwin Bruno Christoffel]]. Levi-Civita,<ref name="Levi-Civita1917">
{{Cite journal|author-link=Tullio Levi-Civita|year=1917|title=Nozione di parallelismo in una varietà qualunque|trans-title=The notion of parallelism on any manifold|url=https://zenodo.org/record/1428456|journal=[[Rendiconti del Circolo Matematico di Palermo]]|language=it|volume=42|pages=173–205|doi=10.1007/BF03014898|jfm=46.1125.02|author-first=Tullio|author-last=Levi-Civita|s2cid=122088291}}
</ref> along with [[Gregorio Ricci-Curbastro]], used Christoffel's symbols<ref>{{cite journal |title=Ueber die Transformation der homogenen Differentialausdrücke zweiten Grades|last=Christoffel|first=Elwin B.|author-link=Elwin Bruno Christoffel|journal=Journal für die reine und angewandte Mathematik|volume=1869|issue=70|pages=46–70|year=1869|url=http://gdz.sub.uni-goettingen.de/dms/load/img/?PPN=GDZPPN002153882&IDDOC=266356|doi=10.1515/crll.1869.70.46|s2cid=122999847}}</ref> to define the notion of [[parallel transport]] and explore the relationship of parallel transport with the [[Riemann curvature tensor|curvature]], thus developing the modern notion of [[holonomy]].<ref>See {{cite book|first=Michael|last=Spivak|author-link=Michael Spivak | title=A Comprehensive introduction to differential geometry (Volume II)|publisher=Publish or Perish Press|year=1999|isbn=0-914098-71-3 |page=238 }}</ref>

In 1869, Christoffel discovered that the components of the intrinsic derivative of a vector field, upon changing the coordinate system, transform as the components of a contravariant vector. This discovery was the real beginning of tensor analysis. 

In 1906, [[L. E. J. Brouwer]] was the first [[mathematician]] to consider the [[parallel transport]] of a [[Euclidean vector|vector]] for the case of 
a space of [[constant curvature]].<ref>
{{Cite journal|author-link=L. E. J. Brouwer|year=1906|title=Het krachtveld der niet-Euclidische, negatief gekromde ruimten|journal=Koninklijke Akademie van Wetenschappen. Verslagen|volume=15|pages=75–94|author-first=L. E. J.|author-last=Brouwer}}
</ref><ref>
{{Cite journal|year=1906|title=The force field of the non-Euclidean spaces with negative curvature|journal=Koninklijke Akademie van Wetenschappen. Proceedings|volume=9|pages=116–133|author-first=L. E. J.|author-last=Brouwer|bibcode=1906KNAB....9..116B}}
</ref>

In 1917, [[Levi-Civita]] pointed out its importance for the case of a [[hypersurface]] immersed in a [[Euclidean space]], i.e., for the case of a [[Riemannian manifold]] embedded in a "larger" ambient space.<ref name="Levi-Civita1917" /> He interpreted the intrinsic derivative in the case of an embedded surface as the tangential component of the usual derivative in the ambient affine space. The Levi-Civita notions of [[intrinsic derivative]] and parallel displacement of a vector along a curve make sense on an abstract Riemannian manifold, even though the original motivation relied on a specific embedding <math>M^n \subset \mathbf{R}^{n(n+1)/2}.</math>

In 1918, independently of Levi-Civita, [[Jan Arnoldus Schouten]] obtained analogous results.<ref>
{{cite journal
 |author-first=Jan Arnoldus
 |author-last=Schouten
 |author-link =Jan Arnoldus Schouten
 |title =Die direkte Analysis zur neueren Relativiteitstheorie
 |journal =Verhandelingen der Koninklijke Akademie van Wetenschappen te Amsterdam
 |volume =12
 |issue=6
 |pages =95
 |year =1918
}}
</ref> In the same year, [[Hermann Weyl]] generalized 
Levi-Civita's results.<ref>
{{cite journal
 |author-first=Hermann 
 |author-last=Weyl
 |author-link =Hermann Weyl
 |title =Gravitation und Elektrizitat
 |journal =Sitzungsberichte Berliner Akademie
 |pages =465–480
 |year =1918
}}
</ref><ref>
{{cite journal
 |author-first=Hermann 
 |author-last=Weyl
 |title =Reine Infinitesimal geometrie
 |journal =[[Mathematische Zeitschrift]]
 |volume =2
 |issue=3–4
 |pages =384–411
 |year =1918
 |doi =10.1007/bf01199420
 |bibcode=1918MatZ....2..384W
 |s2cid=186232500
 |url=https://zenodo.org/record/1447411
}}
</ref>

== Notation ==
* {{math|(''M'', ''g'')}} denotes a [[pseudo-Riemannian manifold]].
* {{math|''TM''}} is the [[tangent bundle]] of {{math|''M''}}.
* {{math|''g''}} is the [[pseudo-Riemannian metric]] of {{math|''M''}}.
* {{math|''X'', ''Y'', ''Z''}} are smooth vector fields on {{math|''M''}}, i. e. smooth [[section (fiber bundle)|section]]s of {{math|''TM''}}.
* {{math|[''X'', ''Y'']}} is the [[Lie bracket of vector fields|Lie bracket]] of {{math|''X''}} and {{math|''Y''}}. It is again a smooth vector field.

The metric {{math|''g''}} can take up to two vectors or vector fields {{math|''X'', ''Y''}} as arguments. In the former case the output is a number, the (pseudo-)[[inner product]] of {{math|''X''}} and {{math|''Y''}}. In the latter case, the inner product of {{math|''X''<sub>''p''</sub>, ''Y''<sub>''p''</sub>}} is taken at all points {{math|''p''}} on the manifold so that {{math|''g''(''X'', ''Y'')}} defines a smooth function on {{math|''M''}}. Vector fields act (by definition) as differential operators on smooth functions. In local coordinates <math>(x_1,\ldots, x_n) </math>, the action reads
: <math>X(f) = X^i\frac{\partial}{\partial x^i}f = X^i\partial_i f</math>
where [[Albert Einstein|Einstein's]] [[Einstein summation convention|summation convention]] is used.

== Formal definition ==
An [[affine connection]] <math>\nabla</math> is called a Levi-Civita connection if
# ''it preserves the metric'', i.e., <math>\nabla g = 0 </math>.
# ''it is [[torsion of connection|torsion]]-free'', i.e., for any vector fields <math>X</math>  and <math>Y</math> we have <math>\nabla_X Y - \nabla_Y X = [X,Y]</math>, where <math>[X, Y]</math> is the [[Lie bracket of vector fields|Lie bracket]] of the [[vector field]]s <math>X</math> and <math>Y</math>.

Condition 1 above is sometimes referred to as [[compatibility with the metric]], and condition 2 is sometimes called symmetry, cf. Do Carmo's text.<ref>{{Cite book |last=Carmo |first=Manfredo Perdigão do |url=https://www.worldcat.org/oclc/24667701 |title=Riemannian geometry |date=1992 |publisher=Birkhäuser |others=Francis J. Flaherty |isbn=0-8176-3490-8 |location=Boston |oclc=24667701}}</ref>

== Fundamental theorem of (pseudo-)Riemannian geometry ==
{{main|Fundamental theorem of Riemannian geometry}}
'''Theorem''' Every pseudo-Riemannian manifold <math>(M,g)</math> has a unique Levi Civita connection <math>\nabla</math>.

''Proof'':<ref>{{cite book|title=Introduction to Riemannian manifolds|author=John M Lee|publisher=Springer-Verlag|year=2018|page=22}}</ref><ref>{{cite book|author=Barrett O'Neill|title=Semi-Riemannian geometry with Applications to relativity|publisher=Academic Press|year=1983|page=61}}</ref>
To prove uniqueness, unravel the definition of the action of a connection on tensors to find 
: <math> X\bigl(g(Y,Z)\bigr) = (\nabla_X g)(Y, Z) + g(\nabla_X Y, Z) + g( Y, \nabla_X Z)</math>.
Hence one can write the condition that <math>\nabla</math> preserves the metric as
: <math> X\bigl(g(Y,Z)\bigr) =  g(\nabla_X Y, Z) + g( Y, \nabla_X Z)</math>.
By the symmetry of <math>g</math>,
: <math> X \bigl(g(Y,Z)\bigr) + Y \bigl(g(Z,X)\bigr) - Z \bigl(g(Y,X)\bigr) = g(\nabla_X Y + \nabla_Y X, Z) + g(\nabla_X Z - \nabla_Z X, Y) + g(\nabla_Y Z - \nabla_Z Y, X)</math>.
By torsion-freeness, the right hand side is therefore equal to
: <math> 2g(\nabla_X Y, Z) - g([X,Y], Z) + g([X,Z], Y) + g([Y,Z], X)</math>.
Thus, the [[Jean-Louis_Koszul|Koszul]] formula
: <math> g(\nabla_X Y, Z) =  \tfrac{1}{2} \Big\{ X \bigl(g(Y,Z)\bigr) + Y \bigl(g(Z,X)\bigr) - Z \bigl(g(X,Y)\bigr) + g([X,Y],Z) - g([Y,Z], X) - g([X,Z], Y) \Big\}</math>
holds. Hence, if a Levi-Civita connection exists, it must be unique, because <math>Z</math> is arbitrary, <math>g</math> is non degenerate, and the right hand side does not depend on <math>\nabla</math>. 

To prove existence, note that for given vector field <math>X</math> and <math>Y</math>, the right hand side of the Koszul expression is linear over smooth functions in the vector field <math>Z</math>, not just real-linear. Hence by the non degeneracy of <math>g</math>, the right hand side uniquely defines some new vector field, which is suggestively denoted <math>\nabla_X Y</math> as in the left hand side. By substituting the Koszul formula, one now checks that for all vector fields <math>X, Y,Z</math> and all functions <math>f</math>,
: <math> g(\nabla_X (Y_1 + Y_2), Z) = g(\nabla_X Y_1, Z) + g(\nabla_X Y_2 , Z) </math>
: <math> g(\nabla_X (f Y), Z) = X(f) g(Y, Z) + f g(\nabla_X Y,Z) </math>
: <math> g(\nabla_X Y, Z) + g(\nabla_X Z, Y) =  X\bigl(g(Y,Z)\bigr)</math>
: <math> g(\nabla_X Y, Z) - g(\nabla_Y X, Z) = g([X,Y], Z). </math>
Hence the Koszul expression does, in fact, define a connection, and this connection is compatible with the metric and is torsion free, i.e. is a Levi-Civita connection.

With minor variation, the same proof shows that there is a unique connection that is compatible with the metric and has prescribed torsion.

== Christoffel symbols ==
Let <math>\nabla</math> be an affine connection on the tangent bundle. Choose local coordinates <math>x^1, \ldots, x^n</math> with coordinate basis vector fields <math>\partial_1, \ldots, \partial_n</math> and write <math>\nabla_j</math> for <math>\nabla_{\partial_j}</math>. The [[Christoffel symbols]] <math>\Gamma^l_{jk}</math> of <math>\nabla</math> with respect to these coordinates are defined as
: <math> \nabla_j\partial_k = \Gamma^l_{jk} \partial_l </math>

The Christoffel symbols conversely define the connection <math>\nabla</math> on the coordinate neighbourhood because
: <math>
\begin{align}
\nabla_X Y &=  \nabla_{X^j\partial_j} (Y^k \partial_k)
           \\&=  X^j\nabla_j(Y^k\partial_k)
           \\ &= X^j\bigl(\partial_j(Y^k)\partial_k + Y^k\nabla_j\partial_k\bigr) 
           \\ &= X^j\bigl(\partial_j(Y^k)\partial_k + Y^k\Gamma^l_{jk}\partial_l\bigr)
           \\ &= X^j\bigl(\partial_j(Y^l) + Y^k\Gamma^l_{jk}\bigr)\partial_l
\end{align}
</math>
that is,
: <math> (\nabla_j Y)^l = \partial_jY^l + \Gamma^l_{jk} Y^k </math>
An affine connection <math>\nabla</math> is compatible with a metric iff
: <math> \partial_i \bigl(g(\partial_j, \partial_k) \bigr)
                         = g(\nabla_i\partial_j, \partial_k) + g(\partial_j, \nabla_i\partial_k)
                         = g(\Gamma^l_{ij}\partial_l, \partial_k) + g(\partial_j, \Gamma_{ik}^l\partial_l)
                         
</math>
i.e., if and only if
: <math> \partial_i g_{jk} = \Gamma^l_{ij}g_{lk} + \Gamma^l_{ik}g_{jl}.</math>
An affine connection {{math|∇}} is torsion free iff
: <math>\nabla_j\partial_k - \nabla_k \partial_j = (\Gamma^l_{jk} - \Gamma^l_{kj})\partial_l = [\partial_j, \partial_k]= 0. </math>
i.e., if and only if 
: <math>\Gamma^l_{jk} = \Gamma^l_{kj}</math>
is symmetric in its lower two indices.

As one checks by taking for <math>X, Y, Z</math>, coordinate vector fields <math>\partial_j, \partial_k, \partial_l</math> (or computes directly), the Koszul expression of the Levi-Civita connection derived above is equivalent to a definition of the Christoffel symbols in terms of the metric as
: <math>\Gamma^l_{jk} = \tfrac{1}{2} g^{lr} \left( \partial _k g_{rj} + \partial _j g_{rk} - \partial _r g_{jk} \right)</math>
where as usual <math>g^{ij}</math> are the coefficients of the dual metric tensor, i.e. the entries of the inverse of the matrix <math>g_{kl}</math>.

== Derivative along curve ==
The Levi-Civita connection (like any affine connection)  also defines a derivative along [[curve]]s, sometimes denoted by {{math|''D''}}.

Given a smooth curve {{math|''γ''}} on {{math|(''M'', ''g'')}} and a [[vector field]] {{math|''V''}} along {{math|''γ''}} its derivative is defined by
: <math>D_tV=\nabla_{\dot\gamma(t)}V.</math>

Formally, {{math|''D''}} is the [[pullback (differential geometry)|pullback connection]] {{math|''γ''*∇}} on the [[pullback bundle]] {{math|''γ''*''TM''}}.

In particular, <math>\dot\gamma(t)</math> is a vector field along the curve {{math|''γ''}} itself. If <math>\nabla_{\dot{\gamma}(t)}\dot{\gamma}(t)</math> vanishes, the curve is called a geodesic of the covariant derivative. Formally, the condition can be restated as the vanishing of the pullback connection applied to <math>\dot\gamma</math>:
: <math>\left(\gamma^*\nabla\right) \dot{\gamma}\equiv 0.</math>

If the covariant derivative is the Levi-Civita connection of a certain metric, then the geodesics for the connection are precisely those [[geodesics]] of the [[Metric tensor|metric]] that are parametrised proportionally to their arc length.

== Parallel transport ==
In general, [[parallel transport]] along a curve with respect to a connection defines [[isomorphism|isomorphisms]] between the tangent spaces at the points of the curve. If the connection is a Levi-Civita connection, then these isomorphisms are [[Orthogonal group|orthogonal]] – that is, they preserve the inner products on the various tangent spaces.

The images below show parallel transport induced by the Levi-Civita connection associated to two different Riemannian metrics on the [[punctured plane]] <math>\mathbf R^2 \backslash \{0,0\}</math>. The curve the parallel transport is done along is the unit circle. In [[Polar coordinate system|polar coordinates]], the metric on the left is the standard [[Riemannian_manifold#Euclidean_space|Euclidean metric]] <math>ds^2 = dx^2 + dy^2 = dr^2 + r^2 d\theta^2</math>, while the metric on the right is <math>ds^2 = dr^2 + d\theta^2</math>. The first metric extends to the entire plane, but the second metric has a singularity at the origin:
: <math>dr = \frac{xdx + ydy}{\sqrt{x^2 + y^2}}</math>
: <math>d\theta = \frac{xdy - ydx}{x^2 + y^2}</math>
: <math>dr^2 + d\theta^2 = \frac{(xdx + ydy)^2}{x^2+y^2} + \frac{(xdy - ydx)^2}{(x^2+y^2)^2}</math>.
{{multiple image
<!-- Essential parameters -->
| align     = center
| direction = horizontal
| caption_align = center
| width     = 200
<!-- Extra parameters -->
| header = Parallel transports on the punctured plane under Levi-Civita connections
| header_align = center
| header_background = 
| footer = 
| footer_align = 
| footer_background = 
| background color =

|image1=Cartesian_transport.gif
|width1=200
|caption1=This transport is given by the metric <math>ds^2 = dr^2 + r^2 d\theta^2</math>.
|alt1=Cartesian transport

|image2=Circle_transport.gif
|width2=200
|caption2=This transport is given by the metric <math>ds^2 = dr^2 + d\theta^2</math>.
|alt2=Polar transport
}}

Warning: This is parallel transport on the punctured plane ''along'' the unit circle, not parallel transport ''on'' the unit circle. Indeed, in the first image, the vectors fall outside of the tangent space to the unit circle.

== Example: the unit sphere in R<sup>3</sup> ==
Let {{math|⟨ , ⟩}} be the usual [[scalar product]] on {{math|'''R'''<sup>3</sup>}}. Let {{math|'''S'''<sup>2</sup>}} be the [[unit sphere]] in {{math|'''R'''<sup>3</sup>}}. The tangent space to {{math|'''S'''<sup>2</sup>}} at a point {{math|''m''}} is naturally identified with the vector subspace of {{math|'''R'''<sup>3</sup>}} consisting of all vectors orthogonal to {{math|''m''}}. It follows that a vector field {{math|''Y''}} on {{math|'''S'''<sup>2</sup>}} can be seen as a map {{math|''Y'' : '''S'''<sup>2</sup> → '''R'''<sup>3</sup>}}, which satisfies
<math display="Block">\bigl\langle Y(m), m\bigr\rangle = 0, \qquad \forall m\in \mathbf{S}^2.</math>

Denote as {{math|''d<sub>m</sub>Y''}} the differential of the map {{math|''Y''}} at the point {{math|''m''}}. Then we have:

{{math theorem|name=Lemma|math_statement= The formula
<math display="block">\left(\nabla_X Y\right)(m) = d_mY(X(m)) + \langle X(m),Y(m)\rangle m</math>
defines an affine connection on {{math|'''S'''<sup>2</sup>}} with vanishing torsion.}}

{{math proof|proof= It is straightforward to prove that {{math|∇}} satisfies the Leibniz identity and is {{math|''C''<sup>∞</sup>('''S'''<sup>2</sup>)}} linear in the first variable. It is also a straightforward computation to show that this connection is torsion free. So all that needs to be proved here is that the formula above produces a vector field tangent to {{math|'''S'''<sup>2</sup>}}. That is, we need to prove that for all {{math|''m''}} in {{math|'''S'''<sup>2</sup>}}
<math display="block">\bigl\langle\left(\nabla_X Y\right)(m),m\bigr\rangle = 0\qquad (1).</math>
Consider the map {{math|''f''}} that sends every {{math|''m''}} in {{math|'''S'''<sup>2</sup>}} to {{math|⟨''Y''(''m''), ''m''⟩}}, which is always 0. The map {{math|''f''}} is constant, hence its differential vanishes. In particular
<math display="block">d_mf(X) = \bigl\langle d_m Y(X),m\bigr\rangle + \bigl\langle Y(m), X(m)\bigr\rangle = 0.</math>
The equation (1) above follows. [[Q.E.D.]]}}

In fact, this connection is the Levi-Civita connection for the metric on {{math|'''S'''<sup>2</sup>}} inherited from {{math|'''R'''<sup>3</sup>}}. Indeed, one can check that this connection preserves the metric.

== Behaviour under conformal rescaling ==
If the metric <math>g</math> in a [[conformal geometry|conformal class]] is replaced by the conformally rescaled metric of the same class <math>\hat g=e^{2\gamma}g</math>, then the Levi-Civita connection transforms according to the rule<ref>{{cite book|author=Arthur Besse|title=Einstein manifolds|year=1987|publisher=Springer|page=58}}</ref>
<math display="block">\widehat\nabla_X Y = \nabla_XY + X(\gamma)Y + Y(\gamma)X - g(X,Y)\mathrm{grad}_g(\gamma).</math>
where <math>\mathrm{grad}_g(\gamma)</math> is the gradient vector field of <math>\gamma</math> i.e. the vector field <math>g</math>-dual to <math>d\gamma</math>, in local coordinates given by <math>g^{ik}(\partial_i \gamma)\partial_k</math>. Indeed, it is trivial to verify that <math>\widehat\nabla</math> is torsion-free.  To verify metricity, assume that <math>g(Y,Y)</math> is constant.  In that case,
<math display="block">\hat g(\widehat\nabla_XY,Y) = X(\gamma)\hat g(Y,Y) = \frac12 X(\hat g(Y,Y)).</math>

As an application, consider again the unit sphere, but this time under [[stereographic projection]], so that the metric (in complex [[Fubini–Study metric|Fubini–Study coordinates]] <math>z,\bar z</math>) is:
<math display="block">g = \frac{4\,dz\,d\bar z}{(1+z\bar z)^2}.</math>
This exhibits the metric of the sphere as conformally flat, with the Euclidean metric <math>dz\,d\bar z</math>, with  <math>\gamma = \ln(2)-\ln (1+z\bar z)</math>.  We have <math>d\gamma = -(1+z\bar z)^{-1}(\bar z\, dz + z\,d{\bar z})</math>, and so
<math display="block">\widehat\nabla_{\partial_z}\partial_z =  -\frac{2\bar z\partial_z}{1+z\bar z}.</math>
With the Euclidean gradient <math>\mathrm{grad}_{Euc}(\gamma) = -(1+z\bar z)^{-1}(\bar z\partial_z + z\partial_{\bar z})</math>, we have
<math display="block">\widehat\nabla_{\partial_z}\partial_{\bar z} =  0.</math>
These relations, together with their complex conjugates, define the Christoffel symbols for the two-sphere.

== See also ==
* [[Weitzenböck connection]]

== Notes ==
{{reflist}}

== References ==
* {{cite book|first=William M.|last=Boothby|author-link=William M. Boothby|title=An introduction to differentiable manifolds and Riemannian geometry |url=https://archive.org/details/introductiontodi0000boot|url-access=registration|publisher=Academic Press|year=1986|isbn=0-12-116052-1}}
* {{cite book|last1=Kobayashi|first1= Shoshichi|author-link1=Shoshichi Kobayashi |last2=Nomizu| first2=Katsumi |author-link2= Katsumi Nomizu |title=Foundations of differential geometry|publisher=John Wiley & Sons|year=1963|isbn=0-470-49647-9}} See Volume I pag. 158

== External links ==
* {{springer|title=Levi-Civita connection|id=p/l058230}}
* [http://mathworld.wolfram.com/Levi-CivitaConnection.html MathWorld: Levi-Civita Connection]
* [https://planetmath.org/levicivitaconnection PlanetMath: Levi-Civita Connection]
* [http://www.map.mpim-bonn.mpg.de/Levi-Civita_connection Levi-Civita connection] at the Manifold Atlas

{{Riemannian geometry}}
{{Manifolds}}
{{Tensors}}

{{DEFAULTSORT:Levi-Civita Connection}}
[[Category:Riemannian geometry]]
[[Category:Connection (mathematics)]]

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Module:Hatnote/styles.css (view source) (template editor protected)
Module:Hatnote list (view source) (template editor protected)
Module:Labelled list hatnote (view source) (template editor protected)
Module:List (view source) (protected)
Module:Multiple image (view source) (template editor protected)
Module:Navbar (view source) (protected)
Module:Navbar/configuration (view source) (protected)
Module:Navbar/styles.css (view source) (protected)
Module:Navbox (view source) (template editor protected)
Module:Navbox/configuration (view source) (template editor protected)
Module:Navbox/styles.css (view source) (template editor protected)
Module:Ns has subpages (view source) (protected)
Module:Pagetype (view source) (protected)
Module:Pagetype/config (view source) (protected)
Module:Pagetype/disambiguation (view source) (protected)
Module:Pagetype/rfd (view source) (protected)
Module:Pagetype/setindex (view source) (protected)
Module:Pagetype/softredirect (view source) (protected)
Module:SDcat (view source) (protected)
Module:String (view source) (protected)
Module:TableTools (view source) (protected)
Module:Template wrapper (view source) (template editor protected)
Module:Unsubst (view source) (protected)
Module:Wikitext Parsing (view source) (protected)
Module:Yesno (view source) (protected)

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Retrieved from "https://en.wikipedia.org/wiki/Levi-Civita_connection"